Nazione¶

Stima downloads dell'app Immuni (vedere link per il metodo utilizzato).

In this jupyter notebook, we'll analyse italian COVID-19 data gathered from Dipartimento di Protezione Civile GitHub repository (link).

We'll consider following compartments

R recovered (total cumulative recovered cases)
D deaths (total cumulative deaths caused by COVID-19)
T confirmed (total cumulative cases)
- infected + recovered + deaths
I infected (current positive cases)
- confirmed - (recovered + deaths)
- H hospitalized + C quarantine, where hospitalized can ben mild or severe (intensive care)

compartments

FIRST ENTRY DATE: 2020-02-24 18:00:00
LAST  ENTRY DATE: 2020-08-31 17:00:00
COVERAGE: 188 days
CURRENT DATE IS:  2020-09-01 12:37:45

Panoramica¶

Variazioni Assolute¶

Variazioni percentuali¶

Letalità e Guarigione¶

!!! PLEASE NOTE !!!

These rates are only useful for SIRD epidemiological model (read here for details) not to define COVID-19 actual rates.

Ospedalizzazione e Isolamento¶

Mobilità¶

R0¶

If the number of Infected $\mathbf{I} \ll \mathbf{N}$ total population (in the SIR model, $\mathbf{S}$usceptible $\rightarrow \mathbf{I}$nfected $ \rightarrow \mathbf{R}$emoved), the basic reproduction number $R_0$ can be calculated as

$$ R_0 \simeq \frac{\beta}{\gamma} $$

and knowing that

$$ d \mathbf{R} = \gamma \mathbf{I} $$$$ d \mathbf{I} = (\beta - \gamma)\mathbf{I} $$

we find that

$$ \gamma = \frac{ d \mathbf{R} }{ \mathbf{I} }$$$$ \beta = \frac{d \mathbf{I}}{\mathbf{I}} + \gamma = \frac{d \mathbf{I} + d \mathbf{R}}{\mathbf{I}} $$

so

$$ R_0 \simeq \frac{ \frac{d \mathbf{I} + d \mathbf{R}}{\mathbf{I}} }{ \frac{ d \mathbf{R} }{ \mathbf{I} } } = \frac{d \mathbf{I} + d \mathbf{R}}{d \mathbf{R}} = \frac{d \mathbf{I}}{d \mathbf{R}} + 1$$

thus since

$$ d \mathbf{R} \geq 0 $$

we can say that

$$ d \mathbf{I} > 0 \Rightarrow R_0 > 1 $$$$ d \mathbf{I} = 0 \Rightarrow R_0 = 1 $$$$ d \mathbf{I} < 0 \Rightarrow R_0 < 1 $$

So when the daily number of new infected is greater than zero, $R_0$ is greater than 1 and could be a sign of an out-of-control epidemy.

When calculated with simplified methods like this (and using ordinary differential equations) $R_0$ is, in fact, simply a threshold, not the average number of secondary infections and can only determine whether an epidemy will die out ($R_0 < 1$) or not ($R_0 > 1$).

Let's try to calculate $R_0$ for Italy, using the iterpolated and smoothed data (to smooth daily fluctuations):

$d \mathbf{R}$ are daily variation of Recovered + Deaths data
$d \mathbf{I}$ is Infected daily variation generated by the sum of daily variation data for Confirmed, Deaths and Recovered
$\mathbf{I}$ is Infected curve generated by the sum of Confirmed, Deaths and Recovered

(read here for details)

R_t bayesiano¶

Italy $R_t$ (effective reproduction number in time $t$) calculated with Bayesian approach (Bettencourt & Ribeiro 2008 and Kevin Systrom 2020).

The vertical dashed line is the day of the national lockdown (dashed) and first lockdown relax (dotted).

Method: italian (PDF)

Look in Italy Regions Overview to view $R_t$ for each region.

Rt come ODDS¶

Effective reproduction number $R_t$ can be considered as an ODD: if for example $R_t=2$, means like in gambling that the likelihood of contagion is "given 2 to 1", so an infected will infect 2 susceptible subjects.

For the effect of the serial interval $\tau$ (onset of symptoms, diagnosis, etc), new cases $k_t$ observed in day $t$ have been infected in $t-\tau$ by the current new cases observed in $t-\tau$ (because old cases are supposed to be hospitalized, isolated, recovered or dead) so, with this simple method, in $t$ we can calculate the $R_{t-\tau}$.

$$ R_{t-\tau} = \frac{k_{t}}{k_{t-\tau}} $$

Work in progress: distribute serial interval $\tau$ as Gamma and new cases $k_t$ as Poisson.